Download Chapter 4 Identities - City Colleges of Chicago

Chapter 4
Identities
4.1 Fundamental Identities and Their Use
4.2 Verifying Trigonometric Identities
4.3 Sum, Difference, and Cofunction Identities
4.4 Double-Angle and Half-Angle Identities
4.5 Product-Sum and Sum-Product Identities
Fundamental Identities and Their Use
Fundamental identities
Evaluating trigonometric identities
Converting to equivalent forms
Fundamental Identities
Evaluating Trigonometric Identities
 Example
Find the other four trigonometric functions of x when
cos x = -4/5 and tan x = 3/4
1
1
5
1
1 4
sec x 


cot x 
 
cos x  4
4
tan x 3 3
5
4
3
 4  3 
sin x  (cos x)(tan x)       
5
 5  4 
1
1
5
csc x 


sin x  3
3
5
Simplifying Trigonometric Expressions
•Claim:
•Proof:
•Claim:
•Proof:
1
2

1

tan
x
2
cos x
1
1  cos 2 x sin 2 x
2

1



tan
x
2
2
2
cos x
cos x
cos x
tan x  cot x
 1  2 cos 2 x
tan x  cot x
sin x cos x
 sin x cos x 


tan x  cot x cos x sin x sin x(cos x)  cos x sin x 




tan x  cot x sin x  cos x sin x(cos x)  sin x  cos x 


cos x sin x
 cos x sin x 
sin 2 x  cos 2 x 1  cos 2 x  cos 2 x
2



1

2
cos
x
2
2
sin x  cos x
1
4.2 Verifying Trigonometric Identities
Verifying identities
Testing identities using a graphing
calculator
Verifying Identities
Verify csc(-x) = -csc x
1
1
1
csc(  x) 


  csc x
sin(  x)  sin x
sin x
Verify tan x sin x + cos x = sec x
sin x
sin 2 x  cos 2 x
1
tan x sin x  cos x 
sin x  cos x 

 sec x
cos x
cos x
cos x
Verifying Identities
Verify right-to-left:
sin x
csc x  cot x 
1  cos x


sin x
sin x 1  cos x 
sin x 1  cos x 



2
1  cos x 1  cos x 1  cos x 
1  cos x
sin x 1  cos x   1  cos x  1  cos x  csc x  cot x
sin 2 x
sin x
sin x sin x




Verifying Identities Using a Calculator
 Graph both sides of the equation in the same
viewing window. If they produce different graphs
they are not identities. If they appear the same
the identity must still be verified.
 Example:
sin x
 csc x
2
1  cos x
4.3 Sum, Difference, and Cofunction
Identities
Sum and difference identities for cosine
Cofunction identities
Sum and difference identities for sine and
tangent
Summary and use
Sum and Difference Identities for Cosine
 cos(x – y) = cos x cos y - sin x sin y
 Claim: cos(p/2 – y) = siny
 Proof:
 cos(p/2 – y) = cos (p/2) cos y + sin(p/2) sin y
= 0 cos y + 1 sin y = sin y
Sum and Difference Formula for Sine and
Tangent
sin (x- y) = sin x cos x + cos x sin y
sin  x  y  sin x cos y  cos x sin y
tan  x  y  


cos x  y  cos x cos y  sin x sin y
sin x cos y cos x sin y

tan x  tan y
cos x cos y cos x cos y

cos x cos y sin x sin y 1  tan x tan y

cos x cos y cos x cos y
Finding Exact Values
 Find the exact value of cos 15º
 Solution:
cos15  cos( 45  30) 
cos 45 cos 30  sin 45 sin 30 


1
3 1 1
3 1
2 3 1


 

4
2 2
2 2 2 2
Double-Angle and Half-Angle Identities
Double-angle identities
Half-angle identities
Double-Angle Identities
1  cos 2 x
1  cos 2 x
2
sin x 
and cos x 
2
2
2
Using Double-Angle Identities
 Example:
Find the exact value of cos 2x if
sin x = 4/5, p/2 < x < p
The reference angle is in the
second quadrant.
a   25  16  3
4
4
sin x  , tan x  
5
3
2
4
 7 
cos 2 x  1  2    
5
 25 
Half-Angle Identities
Using a Half-Angle Identity
 Example: Find cos 165º.
cos165  cos
330
1  cos 330

2
2
3
cos 330  cos 30 
2
3
1
2 3
2
cos165  

2
2
4.5 Product-Sum and Sum-Product
Identities
Product-sum identities
Sum-product identities
Application
Product-Sum Identities
Using Product-Sum Identities
 Example: Evaluate sin 105º sin 15º.
 Solution:
1
sin 105 sin 15  cos105  15  cos105  15 
2
1
1   1  1
cos 90  cos120  0     
2
2   2  4
Sum-Product Identities
Using a Sum-Product Identity
 Example: Write the difference sin 7q – sin 3q as
a product.
 Solution:
sin 7q  sin 3q 
7q  3q
7q  3q
2 cos
sin

2
2
2 cos 5q sin 2q